Locally arc-transitive graphs of valence {3,4}\{3,4\} with trivial edge kernel

In this paper we consider connected locally $G$-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighourhood $\Gamma(u) \cup \Gamma(v)$ is trivial. We find nineteen finitely presented groups with the property that any such group $G$ is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence ${3,4}$ on at most 350 vertices whose automorphism group contains a locally arc-transitive subgroup $G$ with $G_{uv}^{[1]} = 1$

Recent trends and future directions in vertex-transitive graphs

A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade

Combinatorial structures for anonymous database search

This thesis treats a protocol for anonymous database search (or if one prefer, a protocol for user-private information retrieval), that is based on the use of combinatorial configurations. The protocol is called P2P UPIR. It is proved that the (v,k,1)-balanced incomplete block designs (BIBD) and in particular the finite projective planes are optimal configurations for this protocol. The notion of n-anonymity is applied to the configurations for P2P UPIR protocol and the transversal designs are proved to be n-anonymous configurations for P2P UPIR, with respect to the neighborhood points of the points of the configuration. It is proved that to the configurable tuples one can associate a numerical semigroup. This theorem implies results on existence of combinatorial configurations. The proofs are constructive and can be used as algorithms for finding combinatorial configurations. It is also proved that to the triangle-free configurable tuples one can associate a numerical semigroup. This implies results on existence of triangle-free combinatorial configurations

Small triangle-free configurations of points and lines

In this paper we show that all combinatorial triangle-free configurations (v_3) for v (is less than or equal to) 8 are geometrically realizable. We also show that there is a unique smallest astral (18_3) triangle-free configuration, and its Levi graph is the generalized Petersen graph G(18, 5). In addition, we present geometric realizations of the unique flag transitive triangle-free configuration (20_3) and the unique point transitive triangle-free configuration (21_3)